An Introduction to Elliptic Curves
Elliptic Curves and the Abelian Grou
Elliptic Curves are curves of the form
This is called a Weierstrass equation. The rational points on this elliptic curve miraculously create an abelian group under an operation called point addition.
Given two points
and
, we can calculate a third point
on the elliptic curve itself using a simple algorithm:
The images here are from Joseph Silverman's presentation slides from 2006, which are an excellent reference and can be found here.


First, we draw the line
through the points
and
, which intersects the curve at a third point
. We then draw a vertical line passing through
which intersects the curve at a second point we call
.
is the result of
, the point addition of
and
.
We can also add a point
to itself, but how can we do that with infinitely many lines passing through? We simply take the tangent to the curve and find the point
it intersects at before mirroring the point to
.

Rarely, you may find yourself adding
to
, the point directly below
. In this case there is no third point of intersection and we say the result of point addition is
, the point "at infinity".
Last modified 1yr ago